A light-fuelled nanoratchet shifts a coupled chemical equilibrium

Biological molecular machines enable chemical transformations, assembly, replication and motility, but most distinctively drive chemical systems out of-equilibrium to sustain life1,2. In such processes, nanometre-sized machines produce molecular energy carriers by driving endergonic equilibrium reactions. However, transforming the work performed by artificial nanomachines3–5 into chemical energy remains highly challenging. Here, we report a light-fuelled small-molecule ratchet capable of driving a coupled chemical equilibrium energetically uphill. By bridging two imine6–9 macrocycles with a molecular motor10,11, the machine forms crossings and consequently adopts several distinct topologies by either a thermal (temporary bond-dissociation) or photochemical (unidirectional rotation) pathway. While the former will relax the machine towards the global energetic minimum, the latter increases the number of crossings in the system above the equilibrium value. Our approach provides a blueprint for coupling continuous mechanical motion performed by a molecular machine with a chemical transformation to reach an out-of-equilibrium state.


General Information
Chemicals were purchased from commercial sources and used without purification. If not stated otherwise, all reactions were carried out in flame-dried glassware under a nitrogen atmosphere using standard Schlenk techniques. Solutions and reagents were added with nitrogen-flushed disposable syringes/needles. For NMR experiments, solvents were added using glass syringes and stainless steel needles (stored at 80 °C). Analytical thin layer chromatography (TLC) was performed on silica gel 60 G/UV254 aluminium sheets from Merck (0.25 mm). Flash column chromatography was performed on silica gel Davisil LC60A (Merck type 9385, 230-400 mesh) or Reveleris X2 Flash Chromatography system (MPLC) using the indicated solvents. NMR spectra were recorded on a Varian Mercury Plus ( 1 H: 400 MHz, 13 C: 100 MHz), a Varian Unity Plus ( 1 H: 500 MHz, 13 C: 125 MHz) or a Bruker Innova ( 1 H: 600 MHz, 13 C: 151 MHz). Chemical shifts are in parts per million (ppm) relative to TMS. For the calibration of the chemical shift, the residual solvent resonance was used as the internal standard. Data are as follows: chemical shift ( in ppm), multiplicity (br = broad, s = singlet, d = doublet, t = triplet, p = pentet, m = multiplet), coupling constants (J in Hz), integration. High resolution mass spectra (HRMS) were recorded on an LTQ Orbitrap XL. Ion mobility (IM) measurements were performed using a custom drift-tube instrumentation hosted in the Fritz Haber Institute of the Max Planck Society (Berlin, Germany). CD spectra were measured on a Jasco J-815 CD spectrometer. SAXS measurements were performed at the Multipurpose X-ray Instrument for Nanostructure Analysis (MINA) instrument at the University of Groningen. Illuminations were carried out using a UV lamp from Vilber Lourmat (6W, λirr = 365 nm).
The following compounds were prepared according to modified literature procedures: S2 5 , S7-S8 3 , S10-S16 3,4 . Spectra of all compounds described in the synthesis section can be found in the spectra appendix at the end of the Supplementary Information.
The absolute measurement error for NMR spectroscopic measurements involving molecular machine ±n was estimated to be ±3%. The average relative standard deviation of the fit over all individual experiments amounts to < 0.7%. This confirms the proposed kinetic models and the related mechanisms for each individual experiment. The values given in the main manuscript are an average over at least two individual experiments. The relative standard deviation between these individual experiments was estimated to be ±10% for quantum yields; ±15% for intramolecular rate constants; ±15% for experimental Gibbs free energies at 25 °C; ±35% for intermolecular rate constants. For further information see also Supplementary Data Set 1.

Synthesis Synthesis of side-arms
Scheme 1 | Synthesis of side-arms. Synthesis of aldehyde linker S3 (top) and azide linker S5 (bottom). 5 In a flame-dried 3-necked flask equipped with a stirring bar and a reflux condenser, K2CO3 (1.74 g, 12.6 mmol, 3.00 eq.) and 4-hydroxybenzaldehyde (1.03 g, 8.42 mmol, 2.00 eq.) were suspended in dry MeCN (21 mL). Alkyl tosylate S1 (1.50 g, 4.21 mmol, 1.00 eq.) was added and the mixture was heated to 90 °C for 6 h. Subsequently, the mixture was cooled to room temperature and all volatiles were removed under reduced pressure. To the residue, H2O (20 mL) and ethyl acetate (20 mL) were added. The phases were separated and the aqueous layer was extracted with ethyl acetate (3x10 mL) and the combined organic layers were dried over MgSO4. All volatiles were removed under reduced pressure. The title compound S2 was obtained as an off-white solid (1.20 g, 3.92 mmol, 93%) and was used without further purification.

Synthesis of the motor core
Scheme 2 | Synthesis of the motor core.

Cleavage of methyl ethers:
In a flame-dried 3-necked flask, thioxanthon S10 (16.8 g, 62.0 mmol, 1.00 eq.) was dissolved in boiling CH2Cl2 (550 mL). The homogeneous solution was cooled to 0 °C with an ice bath and BBr3 (29.2 mL, 308 mmol, 5.00 eq.) was added slowly. After the addition was complete, the mixture was stirred overnight and allowed to slowly warm to room temperature. Next, the mixture was cooled to 0 °C with an ice bath and MeOH (100 mL) was added slowly. After stirring for 3 h, all volatiles were removed under reduced pressure. The residue was suspended in MeOH (100 mL), which was evaporated again. Next, the crude solid was washed on a glass frit with H2O (ca. 100 mL). Dissolving in acetone (400 mL) and drying over MgSO4 delivered S11 which was used without further purification.

Thioketone formation:
Under a nitrogen atmosphere in a flame-dried 3-necked flask equipped with a stirring bar and a reflux condenser, thioxanthone S12 (2.0 g, 4.0 mmol, 1.0 eq.) was dissolved in dry toluene (20 mL). Lawesson's reagent (5.1 g, 13 mmol, 3.0 eq.) was added and the mixture was heated to 80 °C for 2 h. Subsequently, the mixture was cooled to room temperature and the crude product was purified by flash column chromatography (n-pentane/ethyl acetate = 4:1) and the thioketone was immediately converted further.

Hydrazone formation:
In a round-bottom flask, the thioketone was dissolved in THF (15 mL) and stirred at room temperature. Next, hydrazine (2.1 mL, 4.2 mmol, 10 eq.) was added and the mixture was stirred at room temperature for 2 h. During this time, the color of the reaction mixture changed from green to almost colorless. Then, all volatiles were removed under reduced pressure and the crude product was purified by MPLC (SiO2; n-pentane/ethyl acetate gradient 100:0 ⟶ 80:20). The title compound S13 (2.0 g, 4.0 mmol, 99%) was obtained as a slightly yellow oil which solidified upon standing.
The analytical data is in accordance with the literature.
(R,R,R)-S16      The analytical data is in accordance with the literature.

Postfunctionalization of the motor core
Scheme 3 | Postfunctionalization of the motor core.

General procedure for the formation of (S,R,R)-and (R,R,R)-±n
A stock solution of motor S21 (1.0 mg, 0.60 µmol, 1.0 eq.) in C6D6 was lyophilized in a J. Young NMR tube. Then, PPh3 (0.6 mg, 2.4 mmol, 4.0 eq.) was added and the tube was put under high vacuum for 16 h. After that, the solids were dissolved in dry (distilled from CaH2) and degassed (three freeze-pump-thaw cycles) C6D6 or toluene-d8 (0.6 mL) inside a glovebox and activated molecular sieves (3 Å) were added. The sealed NMR tube was taken out of the glovebox and heated to 60 °C for 7 d in an oil bath under exclusion of light. Subsequently, the molecular sieves were removed inside a glovebox and the sample was used for further experiments without purification.
Since the system is dynamic, the formation of the bridged bis-macrocyclic ±n is concentration dependent. Increasing the concentration leads to formation of an insoluble polymer. The conversion of S21 to ±n was followed using 1 H-NMR spectroscopy by observing the decrease of the aldehyde signal (ca. 9.7 ppm) and increase of imine signals (ca. 8.1 ppm). Typically, bismacrocyclic ±n forms in 90-95%.
Note that PPh3 does not engage in any kind of exchange reaction with the imines. Therefore, no influence of PPh3 on the relaxation rate of wound ±n was observed.
Diastereomer (S,R,R)-±n and its machine-like function was characterized and investigated by HRMS, IMS, CD and NMR spectroscopy, SAXS, and computational studies. Our data supports that the formation of oligomers under our experimental conditions is negligible.

Light-driven winding General illumination conditions
Typically, a J. Young NMR tube containing a 1 mM solution of equilibrated (S,R,R) or (R,R,R)-±n in 0.6 mL C6D6 was illuminated with a lamp for TLC control (365 nm, 6 W) on a shaker plate for 18 min at 8 °C (in a walkable fridge) or room temperature. The distance between lamp and NMR tube was kept constant throughout the experiment. To prevent any significant relaxation, illuminated samples were cooled with a 10 °C acetone bath before the respective measurement.

Stability towards oxygen
Under a nitrogen atmosphere, a sample can be irradiated and relaxed at elevated temperatures several times without noticeable fatigue. Note, that trace amounts of oxygen cannot be fully avoided, which over time has a noticeable impact on the highly dilute samples. First, residual PPh3 acts as an oxygen scavenger and is oxidized to OPPh3, which is then followed by photodecomposition of (S,R,R) or (R,R,R)-±n upon irradiation with UV light of 365 nm. A similar behavior was observed for a comparable system by Giuseppone and co-workers 6 .

Control experiments
To check if bis-macrocyclization of the motor is necessary for the winding mechanism, a 1 mM solution of bis-azide (S,R,R)-S21 in C6D6 was illuminated for 18 min with UV light of 365 nm. No noticeable change in the 1 H-NMR spectrum was observed. Furthermore, we investigated the influence of different macrocyclization stages of (S,R,R)-S21 on the winding process. Therefore, we illuminated (irr = 365 nm, 18 min) a sample that was prepared according to the general procedure for imine macrocyclization after a reaction time of 6.5 h (7% aldehyde conversion) and 27 h (50% aldehyde conversion). This lead to the formation of (S,R,R)-+3 in 0% and 10% yield, respectively ( Supplementary Fig. 21).
We also prepared the n-butyl bis-imine of compound (S,R,R)-S21 in C6D6 (0.6 mL, 1 mM) by in-situ condensation with n-butyl imine (2 eq.). Almost quantitative conversion was observed after three d at 60 °C in presence of 3 Å molecular sieves. Also in this case, no change was observed after illumination with UV light of 365 nm for 18 min (Supplementary Fig. 22).

CD Spectroscopy
In a cuvette, 30 µL of a solution of (R,R,R)-or (S,R,R)-±n (1 mM in C6D6) was diluted with degassed and dry THF to a total volume of 3 mL (c = 10 M ). Benzene absorbs in the deep UV region and forces a cut-off at 270 nm. Spectra of illuminated samples were either recorded after in-situ illumination with a TLC lamp at 365 nm for 5 min (the distance between the cuvette and the lamp was ~2 cm) or by diluting a pre-illuminated sample at PSS with THF. Partial relaxation of the irradiated sample was achieved by heating the sample to 60 °C for 10 min. Subsequent in-situ illumination showed that the process is reversible (Figs. 23 and24). CD spectra for fully protected motors (R,R,R)-S16 and (S,R,R)-S16 were recorded as reference spectra (Supplementary Fig. 25).

NMR Experiments and Kinetic Analysis General
All experiments were conducted on a Varian AVIII 500 NMR spectrometer that was pre-cooled or -warmed to the proper temperature. (S,R,R)-±n samples were typically equilibrated for 5 min inside the instrument until the lock signal reached a constant value. All samples were prepared according to the general procedure and were usually equilibrated at 60 °C. The experimental data were subsequently fitted using COPASI 4.29 7 . In order to obtain a fit that could give a realistic approximation of the irreversible and reversible reactions involved in each experiment, we simulated a reaction compartment of 0.6 mL (to match the volume of the solution of a typical NMR experiment) with concentration of the species involved of 1 mM. In all cases, the time unit used was minutes. The default Levenberg-Marquardt algorithm with a tolerance of 1‧10 −6 implemented in COPASI was used. The initial guess for the kinetic parameter estimation was to consider all species in equilibrium with one another. After every fitting run, visual inspection of the error associated to each kinetic constant provided indication of the relevance of a certain reaction. Kinetic constants with absolute values lower than 10 −6 min −1 were approximated to 0 and the respective reaction deleted in the next iteration. In the following section, the fitting of the experimental values, along with the kinetic model and the associated constants will be provided. For clarity, only a representative example of each dataset is presented. For all the output files of our kinetic simulations see Supplementary Data Set 1.

Relaxation kinetics of a (S,R,R)-±n sample at 10 °C with 20 mol% n-butyl amine
The amine (as a stock solution in deuterated benzene) was added inside a glovebox to a preilluminated (S,R,R)-±n sample (according to the general procedure). The same samples as for the relaxation experiment at 10 °C without nucleophile was used.

Relaxation of a (S,R,R)-±n sample in presence of water
Water (as a stock solution in deuterated THF) was added to a relaxed (S,R,R)-±n sample under an inert atmosphere. The same sample was used for all experiments. Water does not significantly increase the decay rate of (S,R,R)-+3 (k(+3,+2) = 0.2-0.3·10 −3 min −1 ) at 10 °C. However, an illuminated sample (according to the general procedure) containing 20 eq. of water relaxed to isomers (S,R,R)-−1, 0, +1, and +2 at room temperature after 3 d. A similar sample without nucleophile forms almost exclusively (S,R,R)-+2 under the same conditions. Supplementary Fig. 33 | Decay rate of (S,R,R)-+3 in presence of water. The amount of water has no significant effect on the decay rate of (S,R,R)-+3. The first order rate constant k(+3,+2) = 0.2-0.3·10 −3 min −1 was determined by initial slope approximation. C6D6, c = 1 mM, 10 °C, 500 MHz. Supplementary Fig. 34 | 1 H-NMR spectra of irradiated (S,R,R)-±n samples at room temperature after 72 h with and without water. With 20 eq. water (top) and without external nucleophile (bottom). C6D6, c = 1 mM, 500 MHz.

Irradiation kinetics of a (S,R,R)-±n sample at 10 °C without nucleophile
A relaxed (S,R,R)-±n was illuminated at 8 °C inside a walkable fridge according to the general procedure. After each irradiation step, the sample was cooled to 10 °C with an acetone bath to prevent significant relaxation during transportation to the NMR instrument. Further experimental details for the quantum yield determination are given at page 56 of the Supplementary Information. Supplementary Fig. 35 | Irradiation kinetics of a relaxed (S,R,R)-±n sample. Kinetic trace of a representative example with fit (top), NMR stack (middle) and proposed mechanism with average quantum yields (bottom).C6D6, c = 1 mM, 10 °C, 500 MHz.

Mass Spectrometry
Ion mobility (IM) measurements were performed using a custom drift-tube instrumentation hosted in the Fritz Haber Institute of the Max Planck Society (Berlin, Germany) and adapted from a previous design 8 . The instrument is designed around a nanoelectrospray ionization (nESI) source interfaced with a succession of radially-confining entrance funnel, drift tube and exit funnel. This ensemble is prolonged by a quadrupole mass analyzer under high vacuum and ended by an electron multiplier detector (ETP Ion Detect) for ion counting. In practice, samples were diluted to 10 M in acetonitrile and nESI was used to generate ions using a needle voltage of 0.57 kV and a backing pressure of 0.8 bar (N2). The ~160 cm long drift tube was filled with helium buffer gas at a pressure of 4 mbar and subjected to a 2 kV direct current (DC) electric field for mobility separation. Ions were filtered for m/z = 1612 Da, which correspond to the singlyprotonated molecular ion [M+H] + .
Experimental collision cross sections ( DT CCSHe) Ω were determined from the reduced mobility coefficient K0 using Eq. 1 9 after measuring the arrival time distributions (ATD) for DC voltages ranging from 1.3 kV to 2.3 kV. The contribution of each peak was extracted by fitting the ATD using multiple Gaussian functions (OriginPro 2020, OriginLab).
where q is the ion charge, N0 is the standard gas number density, μ is the reduced mass of the iongas colliding partners, kB is the Boltzmann constant and T is the temperature. Supplementary Fig. 36a shows the evolution of the ATD for increasing durations of irradiation at λirr = 365 nm. The transition is characterized by a progressive narrowing of the distribution toward the emergence of a single peak associated with +3 after 15 min. Supplementary Fig. 36b shows the evolution of the ATD for increasing injection voltages applied on the irradiated sample.
The initial single peak corresponding to +3 at low voltage is progressively extinguished in favor of three distinct contributions corresponding to −1, 0 and +1 at high voltages. This latter distribution agrees with the distribution of the non-irradiated sample, thereby validating the reversibility of the reaction pathway.

Computational Analysis
The full thermal rotational path of the motor was probed at the ωB97X-D/def2-TZVP//ωB97X-D/def2-SVP level of theory as implemented in the Gaussian 16, Version B.01 software package 11 . The values of the Gibbs free energies (in kcal/mol) of each species are given in Supplementary Fig.  37. Two different pathways for the thermal helix inversion (THI) were considered (populating either intermediate 2 or 4). To have a better overview of the unidirectionality of the motor, the thermal E-Z barrier (TEZ) was also computed using the broken-symmetry approach at the same level of theory used for calculating the thermal helix inversion step. A difference of more than 6 kcal/mol between the THI and TEZ barriers confirms the unidirectionality of the thermal step of the motor rotation.
The structures of the macrocyclic compounds in different topological isomers were modelled in the respective stable states of the motor core. All the structures were pre-screened using the CREST driver in the xTB software 12-14 using the GFN force field 15 . In this way, the most stable conformers for each structure were picked via the default series of metadynamics and dynamics runs implemented in the driver. The conformers obtained following this procedure were reoptimized at the GFN2-xTB level with very tight optimization criteria. The energy was then computed with a single-point calculation at the M06-2x/def2-SVP level, as implemented in the Gaussian 16, Version B.01 software package 11 . The energy profile of the isomerization process afforded a stepwise increase of the overall energy with a global minimum at the topological isomer 0. The experimentally observed energy differences between the states amount to Gexp (+3,+2) ≥ 2.0 kcal/mol, Gexp (+2,+1) ≥ 2.0 kcal/mol, Gexp (+1,0) = 0.24 kcal/mol and Gexp (−1,0) = 0.39 kcal/mol by assuming that 3% of ±n cannot be reliably detected by 1 H-NMR spectroscopy. These values are comparable with the computed electronic energy differences presented in Fig. 5b of the main text.
The CD spectra of (S,R,R)-±n with progressive number of turns was calculated at the sTDA-xTB level on the optimized structures. The results for the motor core with S-chirality are in Supplementary Fig. 38. The results furnish a qualitative flavor of the CD signs associated to metastable and stable states. From the computations, all the stable states show a positive Cotton effect in the most red-shifted band. The metastable states possess opposite helicity and consequently, opposite sign. Each metastable state was optimized following the same procedure previously discussed for the stable states. Every metastable state generated from a certain stable state was dubbed with an additional ".5" in the name (e.g. +1.5 is generated by photochemical isomerization of +1).

SAXS Measurements
General SAXS measurements were performed at the Multipurpose X-ray Instrument for Nanostructure Analysis (MINA) instrument at the University of Groningen. The instrument is built on a Cu rotating anode high brilliance X-ray source, providing X-ray photons with wavelength of λ = 0.154 nm. The SAXS patterns were recorded using a 2D Vantec500 detector placed 24 cm away from the sample. SAXS 1D profiles were obtained by radially averaging the scattered intensity around the origin of the image (defined by the direct beam position on the detector) using MATLAB. Standard corrections for the detector distortion and sensitivity were applied. The scattering from the buffer solution was subtracted to obtain the neat SAXS signal of the sample. The 1D SAXS profiles are plotted against the modulus of the scattering vector defined as q = 4π sinθ/λ, where θ is half of the scattering angle. The probed scattering angle range was calibrated using known position of diffraction peaks from a standard Silver Behenate sample (NIST).

Sample preparation
A relaxed (S,R,R)-±n sample (1 mM solution in toluene-d8) was contained in a glass capillary of 1.5 mm diameter (wall size of 0.01 mm), flame-sealed to avoid solvent evaporation and placed in the X-ray vacuum chamber to remove air absorption and scattering. The capillary temperature was stabilized at 23 °C. After the measurement, the same solution was illuminated inside the capillary with UV light (irr = 365 nm, 18 min) and immediately measured. To probe reversible conformation change of the molecule, the sample was allowed to relax at 60 °C for 16 h and then measured. We also measured a sample that was pre-illuminated according to the general procedure, which lead to similar results.

Data analysis
As the shape of the nanoratchet in solution is not the one of a simple object (sphere, cylinder, etc.), SAXS profiles were further analyzed to estimate the dimensions of the nanoobject in solution using a model-independent approach. In this case, a generalized Guinier equation was used.
where is the shape factor (0 for sphere, 1 for rod, and 3 for disk), is the radius of gyration and A is a scaling pre-factor that depends on quantities specific of the sample (volume, contrast, and concentration) and specific of the experimental configuration (photon flux, detector sensitivity, and solid angle defined by the detector). The advantage of this approach is that no apriori assumption is made on the shape of the nanoobject in solution, that can be inferred by the fitted value of .
The best fitting curves obtained by this method are in Fig. 2  Our SAXS analysis clearly suggests that the pristine molecules in solution adopt a close-tospheroidal configuration as = 0.19 is close to 0 which is expected for a perfect sphere. On the contrary, winding causes a clear shape change towards an elongated-like conformation as = 0.85 is close to 1 which is expected for a perfect cylinder. In this case, an estimation of the crosssectional radius of gyration for a rod-like conformation (or of the short semi-axis for an ellipsoidal conformation) can be obtained by the cross-sectional Guinier analysis 17 , i.e. fit of the linear part in the plot log( ( )) 2 in the range of points that satisfy the relationship < 1. The value of the slope is related to the cross-sectional radius as = 2 /2. For the irradiated sample (see Supplementary Fig. 40), we get an estimated ~0. 7 .
The larger dimension of the elongated nanoobject can be estimated assuming either an ellipsoidal shape ( 2 = ). We thus estimate ~1.7 nm and ~2.5 nm. These values should be considered with care as they are based on the assumption of a well-defined geometrical shape, but can be nevertheless considered as the lower and an upper limits of the larger axis of the wound nanoobject in solution.

Quantum Yield Determination
The quantum yields for every single winding step of molecular machine (S,R,R)-±n were determined by NMR spectroscopy with ortho-nitrobenzaldehyde (NBA) as an actinometer, following a literature procedure from Ji et al. 18 The following equation was used for the fit: The molar extinction coefficient of NBA is  (365nm) = 265 M -1 cm -1 and its quantum yield is  = 0.5 18 . The light intensity in our irradiation setup was measured at 365 nm and corresponds to a molar photon flux I0 (365nm) = 4.252 mM min -1 ; the path length corresponds to b = 0.027 cm ( Supplementary Fig. 41 left).
The molar extinction coefficient  was determined for the motor core (S,R,R)-S16 ( Supplementary  Fig. 41 right) and corresponds to  (365nm)  3891 M -1 cm -1 . Based on UV/vis and CD spectroscopic experiments, we can assume that  (365nm) is similar for all topological isomers.
Supplementary Fig. 41 | Determination of the light intensitiy (I0), path length (b) and molar extinction coefficient () at 365 nm. The photochemical conversion rates of ortho-nitrobenzaldehyde to ortho-nitrosobenzoic acid at various concentrations give Io and b (left), whereas  was determined by measuring the absorbance of motor core (S,R,R)-S16 at different concentrations (right).
The quantum yield  was then determined by following the temporal change of a fully relaxed sample of (S,R,R)-±n upon illumination with 365 nm by NMR spectroscopy (Supplementary Fig.  35). The kinetic profile was fitted with COPASI 4.29 7 by applying the following equation: Strictly speaking, the quantum yield values provided are apparent quantum yields connecting a stable state ±n of the motor with the successive having an increased crossing number, ±n+1. The metastable form connecting these two states (and the associated ultrafast thermal reaction that populates ±n+1) is neglected because it is impossible to observe under our experimental conditions.

Energetic Considerations of the Winding Mechanism
Our experiments show that AMM ±n works by an energy ratchet mechanism and the winding process is driven by light energy (only the (S,R,R) isomer is considered in this discussion).
Experimentally, the Gibbs free energy ( ) of -and thus strain in -the system increases with increasing amount of crossings ( ), whereas the state with zero crossings ( = 0) has the lowest energy. At thermal equilibrium, three states are populated, namely −1, 0 and +1. States with > +1 or < −1 were not observed.
The system can thermally equilibrate either by intermolecular nucleophile-imine exchange or intramolecular, thermal double bond isomerization. While the rate constant ( ) of the former is independent of , the rate constant of the latter decreases with increasing .
Light-driven winding increases stepwise by +1 and occurs by a photochemical E/Z isomerization, forming a metastable isomer (experimentally not observed) that relaxes by a fast thermal helix inversion (THI). The quantum yield (Ф) of the double bond isomerization decreases with increasing and is therefore dependent on strain in the system. Experimentally, the system reaches up to +3 crossings.
With these observations in mind, we can derive the conditions which limit the number of crossings in our light-driven molecular machine (assuming that no competing nucleophile is present), considering a simple model for our nanoratchet (Supplementary Fig. 42). For a generic molecular motor to operate, an equilibrium between its states (a stable state that was subjected to N turns), + (a metastable state that was subjected to N+1 turns) and + (a stable state that was subjected to N+1 turns) must be established, where ultimately the probabilities to find the system in a certain stable state will be regulated by the Boltzmann equation: + and are the steady-state levels for + and , and + are the free energies of + and , is Boltzmann's constant and is the temperature in Kelvin. This scenario can be described in more detail by considering the equilibria involving these three species on the same potential energy surface and the possibility to populate the respective excited states of species and + , namely * and + * . The diabatic transition from ground to the excited state is regulated by the Bose-Einstein equations for absorption (transition constant ), stimulated and spontaneous emission (transition constant ) 19 . It is known that for Feringa-type molecular motors the population of the productive excited state involved in the isomerization leads to the (almost barrierless) formation of a so-called dark-state * 20,21 . When an equilibrium is reached between the different species and if we consider the two surfaces separated, the ground state will have: The population of * leads to the non-adiabatic formation 20,21 of and + with transition constants and , given the approximation that the directionality of the motion at the excited state does not depend on the state initially populated, but only by the characteristics of * I .
Given these premises, we can for example consider that at the stationary state the probability to photochemically populate the state + starting from via * will be the product of the probabilities associated with each step involving these species II : In this way, we can consider that at the stationary state, the sum of probabilities that lead from and + should equal the sum of the ones that form + from . Thus: 1 = P → + ℎ • P + → + + P → + • P + → + P + → ℎ • P + → + + P + → • P + → + which can be rewritten as: Ф → + describes the quantum yield of the → + process starting from upon light absorption and incorporates both the Bose-Einstein terms and the non-adiabatic terms. We can also explicitly consider the probability of the diabatic transition as proportional to the absorptivity of ( ).
This equation can be zeroed in the following extreme cases, consequently impeding winding and thus the population of + : 1. + ≪ + , hence the metastable state ensuing from the photochemical step ( + ) is more stable than the next "stable" state ( + ) that is generated from the thermal helix inversion (THI) step and/or + → + ≫ + → + , hence further winding is kinetically prevented; 2. Ф → + ≪ Ф + → , given similar molar absorptivities ε and ε , hence the photochemical population of + will not be feasible; 3. ≪ at a given wavelength of irradiation, preventing the excitation of and the formation of + .
III Conditions typical of a photochemically-fueled motor with an E/Z barrier that prevents the thermal population of + from .
High-resolution ESI+ mass spectra of equilibrated (S,R,R)-±n sample ESI+ mass spectra of equilibrated and illuminated (S,R,R)-±n sample equilibrated illuminated